Applications of potential theory in mechanics. Selection of new results

456 Pages·1989·1.587 MB·English

by  Fabrikant V.I.

APPLICATIONS
OF POTENTIAL THEORY
IN MECHANICS
Selection of new results
V.I. Fabrikant
To my parents
(cid:14) (cid:1) (cid:11) (cid:13) (cid:9) (cid:1) (cid:10) (cid:11) (cid:8) (cid:7) (cid:12) (cid:4) (cid:13) (cid:0) (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6) (cid:7) (cid:8) (cid:2) (cid:4)
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PREFACE
It is not easy to find something new in mathematics. It is difficult to
find something new in something very old, like the Potential Theory, which was
studied by the greatest scientists in the past centuries. And it is extremely
difficult to make this find on an elementary level, with no mathematical
apparatus involved which would be considered new even in the times of, say,
Poisson. This is exactly what the author claims to have done in this book: a
new and elementary method is described for solving mixed boundary value
problems, and their applications in engineering. The method can solve
non-axisymmetric problems as easily as axisymmetric ones, exactly and in closed
form. It enables us to treat analytically non-classical domains. The major
achievements of the method comprise the derivation of explicit and elementary
expressions for the Green’s functions, related to a penny-shaped crack and a
circular punch, development of the Saint-Venant type theory of contact and crack
problems for general domains, and investigation of various interactions between
cracks, punches, and external loadings. The method also provides, as a bonus, a
tool for exact evaluation of various two-dimensional integrals involving distances
between two or more points. It is believed that majority of these results are
beyond the reach of existing methods.
For over twenty years, since being a graduate student in Moscow, the
author had been perplexed by an inconsistency between various solutions to the
problems in Potential Theory and the way those solutions have been obtained,
namely, that the solution was quite elementary, while the apparatus used was
very complicated, involving various integral transforms or special functions
expansions, which are beyond the comprehension of an ordinary engineer. The
author’s search for a new method was based on the conviction that an
elementary result should be obtainable by elementary means. The method has
been found and is presented here in detail. One may just wonder why the
method was not discovered at least a century ago.
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The book is addressed to a wide audience ranging from engineers, involved
in elastic stress analysis, to mathematical physicists and pure mathematicians.
While an engineer can find in the book some elementary, ready to use formulae
for solving various practical problems, a mathematical physicist might become
interested in new applications of the mathematical apparatus presented, and a pure
mathematician might interpret some of the results in terms of fractional calculus,
investigate the group properties of the operators used, or, having noticed the fact
that no attention is paid in the book to a rigorous foundation of the method,
might wish to remedy the situation. Due to the mathematical analogy between
mixed boundary value problems in elasticity and in other branches of engineering
science, the book should be of interest to specialists in electromagnetics,
acoustics, diffusion, fluid mechanics, etc. Though several such applications have
been published by the author, the space considerations did not allow us to
include them, but references are given at appropriate places in the book.
The book is accessible to anyone with a background in university
undergraduate calculus, but should be of interest to established scientists as well.
Though the method is elementary, the transformations involved are sometimes very
non-trivial and cumbersome, while the final result is usually very simple. The
reader who is interested only in application of the general results to his/hers
particular problems may skip the long derivations and use the final formulae
which require little effort. The reader who wants to master the method in order
to solve new problems has to repeat the derivations which are given in sufficient
detail. The exercises are important in this regard. They vary from very simple
to quite difficult. Some can be used as a subject for a graduate degree thesis.
The book is based entirely on the author’s results, and this is why the
work of other scientists is mentioned only when such a quotation is inevitable
for some reason, like numerical data needed to verify the accuracy of
approximate results, comparison with existing results, or pointing out some errors
in publications. There are several books and review articles presenting an
adequate account of the state-of-the-art in the field. Appropriate references are
given for the reader’s convenience. The purpose of this book was neither to
repeat nor to compete with them.
The development of the method can by no means be considered completed,
this is just a beginning. The results presented in the book may be compared to
the tip of an iceberg, taking into consideration numerous applications which are
still to come. The solution of fundamental problems in a simple form enables
us to consider various more complicated problems which were not even attempted
before. The method can be expanded to spherical, toroidal and other systems of
coordinates, so that more complex geometries may benefit from it. The method
proved useful in the generalized potential theory as well. Some of these results,
though already published by the author, could not be included in the book due
to severe restrictions on the book volume.
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For the reader’s convenience, it was attempted to make each chapter (and
section, wherever possible) self-contained. The reader can skip several sections
and continue reading, without loosing the ability to understand material. On the
other hand, this resulted in repetition of some definitions and descriptions. The
author thinks that the additional convenience is worth several extra pages in the
book.
The author is grateful to Professor J.R. Barber from Michigan, Professor
B. Noble from England, and Professor J.R. Rice from Harvard who agreed to
read the manuscript and expressed their opinion.
The book contains so much new material that some misprints and errors are
inevitable, though every effort was made to eliminate them. The author would
be grateful for every communication in this regard. All the readers’ comments
are welcome. The address is
V.I. Fabrikant
prisoner #167932D
Archambault jail
Ste-Anne-des-Plaines
Quebec, Canada J0N 1H0
CONTENTS
INTRODUCTION 7
CHAPTER 1. DESCRIPTION OF THE NEW METHOD
1.1 Integral representation for the reciprocal
of the distance between two points 15
1.2 Properties of the L-operators 22
1.3 Some further integral representations 25
1.4 Internal mixed boundary value problem for a half-space 31
1.5 External mixed boundary value problem for a half-space 43
1.6 Some fundamental integrals 63
CHAPTER 2. MIXED BOUNDARY VALUE PROBLEMS
IN ELASTICITY
2.1 General solution 73
2.2 Point force solutions 77
2.3 Internal mixed problem of type I 81
2.4 External mixed problem of type I 92
2.5 Integral representation for q2/R3 101
2.6 Internal mixed problem of type II 105
2.7 External mixed problem of type II 111
2.8 Inverse crack problem in elasticity 129
CHAPTER 3. MIXED-MIXED BOUNDARY VALUE PROBLEMS
3.1 General formulation of the problem 142
3.2 Internal axisymmetric mixed-mixed problem 145
3.3 External axisymmetric mixed-mixed problem 171
3.4 Generalization for non-homogeneous half-space 177
3.5 Effect of a shearing force and a tilting moment
on a bonded circular punch 182
3.6 Non-axisymmetric internal mixed-mixed problem 188
3.7 Non-axisymmetric external mixed-mixed problem 200
Appendix A3.1 207
Appendix A3.2 211
Appendix A3.3 215
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CHAPTER 4. APPLICATIONS IN FRACTURE MECHANICS
4.1 Flat crack under arbitrary normal loading 220
4.2 Point force loading of a penny-shaped crack 227
4.3 Concentrated load outside a circular crack 229
4.4 Plane crack under arbitrary shear loading 231
4.5 Penny-shaped crack under uniform pressure 243
4.6 Penny-shaped crack under uniform shear loading 248
4.7 Asymptotic behavior of stresses and displacements
near the crack rim 252
4.8 Flat crack of general shape 259
4.9 General crack under uniform shear 273
4.10 Close interaction of pressurized coplanar circular cracks 275
4.11 Close interaction of coplanar circular cracks
under shear loading 292
Appendix A4.1 307
Appendix A4.2 311
Appendix A4.3 312
Appendix A4.4 317
CHAPTER 5. APPLICATION TO CONTACT PROBLEMS
5.1 Contact problem for a smooth punch. 324
5.2 Flat centrally loaded circular punch 328
5.3 Inclined circular punch on an elastic half-space 332
5.4 Flat punch of arbitrary planform under the action
of a normal centrally applied force 336
5.5 Inclined flat punch of general shape 347
5.6 Curved punch of general planform 367
5.7 Flat flexible punch of general planform
under shifting load 387
5.8 Reissner-Sagoci problem for general domains 393
5.9 Interaction between punches subjected to normal pressure 402
5.10 Interaction between flexible punches under shifting loading 412
5.11 Contact problem for a rough punch 422 Appendix A5.1
425
REFERENCES 439
SUBJECT INDEX 447
INTRODUCTION
A short survey of methods for solving mixed boundary value problems of
potential theory is given, and some of their limitations are pointed out. The
necessity for developing a new method is justified. A concise description of
each chapter is given for the reader’s convenience.
Various applications of potential theory in electrostatics, heat transfer,
elasticity, diffusion, and other branches of engineering science are well known,
and have attracted significant attention of scientists like Laplace, Poisson, Green,
Beltrami, Kirchhoff, Lord Kelvin, Hobson, and others who made a significant
contribution to the field during past centuries. The boundary value problems
with mixed conditions are the most difficult to solve, and, at the same time,
they are among the most important in various engineering applications. Recall
the celebrated problem of an electrified circular disc. It is next to impossible
even to name every scientist who solved the problem by one method or another.
We can specify two categories of methods. The first one requires construction
of the Green’s function, after which each particular solution can be presented in
quadratures. The second category encompasses various integral transform methods.
Let us briefly discuss both categories.
Hobson (1899) has constructed the Green’s function for a circular disc and
a spherical bowl using a method due to Sommerfeld. He has used toroidal
coordinates t , s , f (in our notation), which are related to the cartesian x, y, and
z as follows:
asinht cosf asinht sinf asins
x = , y = , z = .
cosht - coss cosht - coss cosht - coss
Here a is the disc radius. The potential function V at the point (t ,s ,f )
0 0 0
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8 INTRODUCTION
external to the disc, which on the disc takes the values v(t ,f ), can be expressed
(cid:236) (1 + cosht ) cos(s /2)
1 (cid:243) (cid:243) 0
V(t ,s ,f ) = (cid:237)
0 0 0 p 2 ı ı (cid:238) 2aR[cosh2(a /2) - sin2(s /2)]1/2
S 0
(1 - coss )1/2 (cid:252)
z
0
+ tan-1 (cid:253) v(t ,f ) dS, (0.1)
R3 (cosha + coss )1/2(cid:254)
0
where R is the distance between the points (t ,s ,f ) and (t ,s ,f ), and a is
0 0 0
defined by cosha = cosht cosht - sinht sinht cos(f - f ). The practical value of
0 0 0
expression (0.1) is quite limited, since there seems to be no way to evaluate the
integral involved, even in the simplest case when v(t ,f ) is constant. Hobson had
to use a very ingenious method in order to find the potential function for
v=const and v=m x, m =const. On the other hand, it turns out that the integral in
(0.1) is computable in elementary functions for any polynomial v (in Cartesian
coordinates). This was one of the reasons that prompted the author to look for
an alternative approach, which would be as general as (0.1), and, on the other
hand, would allow elementary and straightforward computation of the integrals
involved.
Consideration of the mathematically equivalent problem of a circular punch
pressed against an elastic half-space leads to the integral equation (Galin, 1953)
2p a s (r ,f )
(cid:243) (cid:243) 0 0
w (r ,f ) = H r dr df . (0.2)
ı ı R 0 0 0
0 0
Here a is the punch radius, H is an elastic constant, w is the normal
displacement under the punch (a known function), s stands for the pressure
exerted by the punch (an unknown function), and R is the distance between the
points (r ,f ) and (r ,f ). Leonov (1953) has obtained a closed form exact
0 0
solution of integral equation (0.2) by a very ingenious method. His result reads
in our notation
(cid:236) 2p a w (r ,f )
1 (cid:243) (cid:243) 0 0
s (r ,f ) = - (cid:237) D r dr df
4p 2H (cid:238) ı ı R 0 0 0
0 0
INTRODUCTION 9
2p a w (r ,f ) (cid:252)
2 (cid:243) (cid:243) Ø R R ø 0 0
+ - tan-1( ) r dr df (cid:253) . (0.3)
p ı ı º h h ß R3 0 0 0(cid:254)
0 0
Here h =[(a2- r 2)(a2- r 2)]1/2/a and D is the two-dimensional Laplace operator. One
0
can observe the same handicap in (0.3): difficulty to evaluate the integrals
directly, even in the simplest case when w is constant. Again, it is clear that
the integrals are computable in elementary functions for any prescribed polynomial
displacement. This indicates a gap in our knowledge which needs to be filled.
The integral transform method, involving dual integral equations, was
originated, probably by Weber (1873) and Beltrami (1881), and continued by
Busbridge (1938) and others. Significant achievements in the systematic
application of the method to various problems belong to Sneddon. The reader is
referred to the books by Sneddon (1951, 1966) and Ufliand (1967) for additional
references. Some quite remarkable results were obtained by Ufliand (1977).
Despite this success, it has always been the author’s conviction that our use of
integral transforms generally indicates our inability to solve problems directly. To
this end, two illustrative examples are presented.
Here is how the problem of a circular disc, charged to a potential
v =const, is solved by the dual integral equation method. It is necessary to find
0
a harmonic function V, vanishing at infinity, and subjected to the mixed
boundary conditions on the plane z=0:
¶ V
V = v , for r £ a; = 0, for r >a. (0.4)
0 ¶ z
The solution is presented in the form
¥
(cid:243) dt
V(r ,z) = A(t)e-tzJ (tr ) . (0.5)
ı 0 t
0
Here J is the Bessel function of zero order, and A(t) is the as yet unknown
0
function which should be chosen to satisfy (0.4). Substitution of the boundary
conditions (0.4) in (0.5) leads to the dual integral equations
10 INTRODUCTION
¥
(cid:243) dt
A(t)J (tr ) = v , for 0£ r £ a;
ı 0 t 0
0
¥
(cid:243)
A(t)J (tr )dt = 0, for r >a. (0.6)
ı 0
0
By using the discontinuous Weber-Schafheitlin integrals, one can deduce that
2
A(t) = v sin(at). (0.7)
p
0
The solution (0.5) can now be rewritten as
¥
2 (cid:243) dt
V(r ,z) = v e-tzsin(at) J (tr ) , for z‡ 0, (0.8)
p 0ı 0 t
0
and the charge density s over the disc is given by
v ¥
0 (cid:243)
s (r ) = J (tr ) sin(at) dt. (0.9)
p 2 ı 0
0
Both integrals (0.8) and (0.9) are computable in elementary functions. For
example, the last integral yields
v
0
s (r ) = . (0.10)
p 2(a2 - r 2)1/2
There seems to be a discrepancy between the simplicity of the final result and
the apparatus used to obtain it. The general idea, that an elementary result
should be obtained by elementary means, calls for a search for a new and
elementary approach.
The second example comes from consideration of the simplest case of a
penny-shaped crack of radius a, subjected to axisymmetric pressure p(r ). The
corresponding potential function f can be found in (Kassir and Sih, 1975) as
follows: